Решить уравнение: -sin(x/2)+cos(x/4)=0

Question

Решить уравнение:

-sin(x/2)+cos(x/4)=0

Answer 1

-sin(x/2) + cos(x/4) = 0
Разложим sin(x/2) по формуле удвоенного аргумента
-2cos(x/4)sin(x/4) + cos(x/4) = 0
cos(x/4)[-2sin(x/4) + 1] = 0
cos(x/4) = 0
x/4 = π/2 + πn, n ∈ Z
x = 2π + 4πn, n ∈ Z
-2sin(x/4) = -1
sin(x/4) = 1/2
x/4 = (-1)ⁿπ/6 + πk, k ∈ Z
x = (-1)ⁿ2π/3 + 4πk, k ∈ Z
Ответ: x = 2π + 4πn, n ∈ Z; (-1)ⁿ2π/3 + 4πk, k ∈ Z.

Answer 2

-sin(x/2)+cos(x/4)=0
-2sin(x/4)cos(x/4)+cos(x/4)=0
-cos(x/4)*(2sin(x/4)-1)=0
cosx/4=0⇒x/4=π/2+2πk⇒x=2π+8πk,k∈z
sinx/4=1/2
x/4=π/6+2πk,k∈z U x=5π/6+2πk,k∈z
x=2π/3+8πk,k∈z U x=10π/3+8πk,k∈z